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We study the topological properties of random closed curves on the orientable surface $S$ of negative Euler characteristic. Let $\gamma_{n}$ denote the conjugate class of $n^{th}$ steps of a simple random walk on the Cayley graph driven by a non-elementary measure on the standard generating set, with probability converging to $1 When$ $n goes to infinity, (1) the length of$\gamma_{n}$is minimized. (2) the number of self-intersections of$\gamma_{n}$is on the order of$n^{2}$and (3)$S$is punctured and the distribution is If uniform, if the minimum order$\gamma_{n}$of the cover allows a simple lift (called$\textit{simple lift}$of$\gamma_{n}$), then at least$ It grows as n/\log(n)$and is on the order of$n$at most. It also shows that these properties are$\textit{generic}$. This means that the proportion of elements of spheres of radius$n$in the Cayley graph they hold converges to$1$as$n$goes. infinite. The resulting simple lifting bound for the randomly chosen curve significantly improves the previously known best range, which was of the order of$\log^{(1/3)}n\$. increase. As an application, we give the extension of the general point-push pseudo-Anosov homeomorphism relatively sharp upper and lower bounds on the number of self-intersections of its defining curve, and also give an upper bound on the random simple lifting order. A curve with a better number of points of intersection than the general curve boundary.

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